2.2. LAPLACE’S EQUATION 29 c. Local estimates for harmonic functions. Now we employ the mean- value formulas to derive careful estimates on the various partial derivatives of a harmonic function. The precise structure of these estimates will be needed below, when we prove analyticity. THEOREM 7 (Estimates on derivatives). Assume u is harmonic in U. Then (18) |Dαu(x0)| ≤ Ck rn+k u L1(B(x0,r)) for each ball B(x0,r) ⊂ U and each multiindex α of order |α| = k. Here (19) C0 = 1 α(n) , Ck = (2n+1nk)k α(n) (k = 1,... ). Proof. 1. We establish (18), (19) by induction on k, the case k = 0 being immediate from the mean-value formula (16). For k = 1, we note upon differentiating Laplace’s equation that uxi (i = 1,...,n) is harmonic. Con- sequently (20) |uxi (x0)| = − B(x0,r/2) uxi dx = 2n α(n)rn ∂B(x0,r/2) uνi dS ≤ 2n r u L∞(∂B(x0, r 2 )). Now if x ∈ ∂B(x0,r/2), then B(x, r/2) ⊂ B(x0,r) ⊂ U, and so |u(x)| ≤ 1 α(n) 2 r n u L1(B(x0,r)) by (18), (19) for k = 0. Combining the inequalities above, we deduce |Dαu(x0)| ≤ 2n+1n α(n) 1 rn+1 u L1(B(x0,r)) if |α| = 1. This verifies (18), (19) for k = 1. 2. Assume now k ≥ 2 and (18), (19) are valid for all balls in U and each multiindex of order less than or equal to k − 1. Fix B(x0,r) ⊂ U and let α

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